Symmetry group analysis of Benney system and an application for shallow-water equations

In this study we analyze symmetry group properties of the Benney system in the Eulerian description, which is in the form of the system of the coupled nonlinear integro-differential equations. We, first, find symmetry groups and obtain reduced forms, and then seek some similarity solutions to the reduced forms of the Benney equations. In addition, it is shown that one may transform solutions of the reduced forms of the Benney system into solutions of the reduced form of the one-dimensional shallow-water equations.